You must be the account owner of the HSA. You may not need to be the primary insured, if you are not a tax dependent but are covered by a parents HDHP.
If your employer won’t allow you to join their HSA and receive free employer money if you aren’t the primary insured, but you are eligible to make HSA contributions, you could open a personal HSA at any convenient bank and make tax deductible contributions from your take home pay. You won’t get free money from your employer but you would get the tax savings and long term growth from your contributions.
Or, join your employer’s HDHP and have your parents take you off of their plan. You becoming eligible for your own insurance would generally be a qualifying event that allows your parents to change their coverage even outside the normal open enrollment period.
